Question

Which point would be a solution to the system of linear inequalities shown below?

Answer 1

(-2, 10)

Explanation:

You want the point that would be a solution to the inequalities ...

• y ≥ 5/2x +2
• y ≥ -4x -7

Graph

It can be useful to graph the inequalities, or develop a mental picture of what the graph would look like. Both boundary line slopes are fairly steep, and the lines cross in the third quadrant. The V-shaped space above that intersection is the solution space.

The attachment shows the point (-2, 10) is a solution.

Try the answers

From the shape and location of the solution space, we can eliminate the choices ...

(-8, 2) — too close to the x-axis in the far left part of the 2nd quadrant

(10, -3) — no part of the 4th quadrant is in the solution space

General form

It can work nicely to rewrite the inequalities as a comparison to zero.

5x -2y +4 ≤ 0 . . . . . the first inequality in general form

point (-2, 10): 5(-2) -2(10) +4 = -10 -20 +4 = -26 ≤ 0 . . . a solution

point (4, 9): 5(4) -2(9) +4 = 20 -18 +4 = 6 > 0 . . . . . . . not a solution

4x +y +7 ≥ 0 . . . . . . the second inequality in general form

point (-2, 10): 4(-2) +(10) +7 = -8 +10 +7 = 9 ≥ 0 . . . . . . a solution

point (4, 9): don't need to test (already known not a solution)

Point (-2, 10) is a solution.

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Additional comment

We chose the use of "general form" inequalities for evaluating answer choices because ...

• the arithmetic is mainly with positive integers (no fractions)
• the comparison to zero does not require a lot of mental effort

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